Project number		Stand-alone Projects  P20229
Title		Combinatorial Set Theory of the Real Line
Principal investigator		MILDENBERGER Heike
Approval date		01.10.2007
University / Research institution		Kurt Gödel Center for Mathematical Logic, Universität Wien
Scientific field(s)
Keywords		Forcing, Ultrafilter, Iterationstechniken, Kohärenzklassen, Kardinalzahlcharakteristiken, Karoprinzipien
Homepage		http://www.logic.univie.ac.at/~heike

We propose to investigate combinatorial properties in the set theory of the real line, a subfield of mathematical logic. The proposed research has applications to open problems in topology, algebra and combinatorics of Aleph_1. Specific applications concern: the number of near-coherence classes of ultrafilters, the existence of subgroups of the Baer-Specker group that are bounded in one dimension but unbounded in a higher dimension, and the connections between "guessing principles'' and the existence of Souslin trees.

In the proposed field, independence of ZFC is very likely. Therefore the main part of the proposed work is to develop forcing techniques. I also want to emphasize combinatorial methods in the analysis of existing notions of forcing with respect to new properties. In order to determine whether a given forcing extension has a property, the forcing notion might need to be further specified, because the forcing notion creates a whole class of extension models. This class is given axiomatically and hence Gödel's incompleteness theorem applies and the question whether the forcing notion forces a statement can be undecided. The art is to combine forcing technology with the right tools from combinatorial set theory.

In the proposed work, cardinal characteristics of the continuum describe important combinatorial features of the ZFC models studied. A cardinal characteristic of the continuum locates the smallest size of a set having a property that is typically not exhibited by any countable set, but is exhibited by at least one set of size of the cardinality of the continuum. Usually the value of a cardinal characteristic is not determined by ZFC.